Question

If \(\sin\theta = \frac{11}{61}\), find the value of \(\cos\theta\) using trigonometric identity.

Hint:

Knowing Pythagorean triples can be very helpful. In this problem, \(\sin\theta = \frac{11}{61} = \frac{\text{Opposite}}{\text{Hypotenuse}}\). If you recognize the triple (11, 60, 61), you can immediately deduce that the adjacent side must be 60, so \(\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{60}{61}\).

Answer 1

Step 1: Understanding the Concept:
We need to use the fundamental Pythagorean trigonometric identity that relates the sine and cosine of an angle.

Step 2: Key Formula or Approach:
The Pythagorean identity is: \[ \sin^2\theta + \cos^2\theta = 1 \] We can rearrange this to solve for \(\cos\theta\): \[ \cos^2\theta = 1 - \sin^2\theta \] \[ \cos\theta = \sqrt{1 - \sin^2\theta} \]

Step 3: Detailed Explanation:
Given: \[ \sin\theta = \frac{11}{61} \] Using the identity \( \sin^2\theta + \cos^2\theta = 1 \): \[ \left(\frac{11}{61}\right)^2 + \cos^2\theta = 1 \] \[ \frac{11^2}{61^2} + \cos^2\theta = 1 \] \[ \frac{121}{3721} + \cos^2\theta = 1 \] Now, solve for \(\cos^2\theta\): \[ \cos^2\theta = 1 - \frac{121}{3721} \] \[ \cos^2\theta = \frac{3721 - 121}{3721} \] \[ \cos^2\theta = \frac{3600}{3721} \] Take the square root of both sides. Since it is not specified, we assume \(\theta\) is in the first quadrant where cosine is positive. \[ \cos\theta = \sqrt{\frac{3600}{3721}} \] \[ \cos\theta = \frac{\sqrt{3600}}{\sqrt{3721}} \] \[ \cos\theta = \frac{60}{61} \]

Step 4: Final Answer:
The value of \(\cos\theta\) is \( \frac{60}{61} \).